Computational techniques for modeling the structural response of complex structures under load have found widespread use in industry and academia. These techniques enable a relatively simple structure or component to be designed such that it performs to specification during its operation or deployment. However, in order to enable modeling of complex ‘real-world’ structures or components, it is common to make one or more simplifying assumptions regarding the geometry of the structure, the nature of the loading applied to the structure, and/or the material properties of the structure. Unfortunately, assumptions of this nature can cause the computational models to produce spurious results by virtue of failing to capture important ‘real-world’ structural behaviour. This is particularly disadvantageous in cases where the real-world behaviours have a critical effect on the performance of the structure or component. For example, a real-world structure may fail at lower loads than predicted for the modeled structure, or the failure mode of the real-world structure may not correspond to that determined for the modeled structure. One solution to this problem is to account for the real-world factors by incorporating a safety factor into the structure; however, this can result in components with non-optimum material usage, weight and cost.
One computational technique which has found widespread use is the Finite Element Method (FEM) which is ubiquitous in the field of stress analysis of complex structures. FEM is a numerical technique for finding approximate solutions to partial differential equations (PDEs) in complex systems and involves discretisation of the structure (or ‘domain’) into a plurality of elements (forming a ‘mesh’) defined by points in space (‘nodes’). One or more boundary conditions which define the behaviour of the structure at its boundaries are applied to the mesh to model applied loads, moments or displacements. In combination, the mesh and its associated boundary conditions form a ‘model’ which serves as input to the FEM.
The discretisation process for all elements in the domain and the application of the one or more boundary conditions results in matrix equation of finite dimension, the solution of which will approximately solve the original PDEs describing the physical behaviour of the structure under analysis.
In relation to structural analysis, FEM can be broadly categorised as either linear FEM or non-linear FEM. Linear FEM is computationally efficient but does not capture a number of real-world behaviours (e.g. geometric non-linearity, material non-linearity and non-linear boundary conditions). In contrast, non-linear FEM is capable of capturing the various non-linear behaviours of a structure but is often prohibitively expensive from a computational perspective.
For structures which are substantially planar in shape, application of out-of-plane forces can often lead to large displacements and thus a highly non-linear geometric response (e.g. buckling). Generally, such phenomena cannot be captured by linear FEM where large displacements are ignored. However, the solution of non-linear FEM is computationally expensive in comparison to that of linear FEM so it is desirable to develop techniques this alleviate this drawback. One technique which has found widespread use is a ‘global-local’ approach, whereby linear FEM is used to model the linear response of a global structure (GFEM), and the results of the linear GFEM are applied a local model (representing a part of the global structure) for subsequent non-linear analysis (LFEM). According to this approach, solution of a non-linear problem is required for only the local model, thereby providing a more tractable solution from a computational perspective.
When employing a global-local approach, it is important to define an effective post-processing scheme for extraction of the results of the linear GFEM analysis and transferal of the same to the non-linear LFEM analysis (otherwise known as ‘mapping’). Ideally, the chosen mapping scheme should ensure that the results of the non-linear LFEM are as close as possible to those that would be obtained if the non-linear analysis was performed on the GFEM, and to those of the real-world behavior of the structure being analyzed.